Why Vectors Aren't Just Arrows | Foundational Mathematics for Science and Engineering | Ep.9

Описание: In this video, we develop the mathematical framework of linear vector spaces, which underpins modern physics, especially quantum mechanics and advanced classical theory. We introduce vectors, inner products, bases, operators, and eigenvalue problems with a strong focus on structure and physical meaning.

In this video, you'll learn:
Definition and axioms of linear vector spaces
Scalars, fields, and vector components
Linear independence, basis, and dimension
Inner product spaces and orthonormal bases
Gram–Schmidt orthogonalization
Linear operators and matrix representations
Hermitian and unitary operators
Eigenvalues, eigenvectors, and degeneracy
Cayley–Hamilton theorem
Function spaces and Hilbert spaces
Fourier series and Fourier transforms
Dirac delta function
Introduction to tensors and tensor transformations

Here is the notes for this chapter
https://drive.google.com/file/d/1MEIe...

Here are the Timestamps
0:00 Table of Contents
1:51 Basics
10:45 Inner Product Spaces
24:28 Linear Operators
30:32 Some Advanced Topics
33:11 Eigenvalue Problem
48:18 Application of Eigenvalue Problem
54:22 Function Spaces
1:08:58 Terminology
1:11:28 Introduction to Tensors

Whether you're a high school student, a university student, or just curious, this series is your launchpad to strengthen the mathematics background for science and engineering.

Drop your thoughts or questions in the comments — and stay tuned for deep dives into each chapter, with real data, visual simulations, and critical thinking.

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